Chapter 3. Quantum physics from Planck and
Einstein to Bohr, Heisenberg, de Broglie, and Schrödinger

Physicists measure the spectrum (the intensity of light as a function of
wavelength, or color) of a light source in a spectrometer. The figure below
shows a schematic drawing of a simple prism spectrometer. White light comes
in from the left and the prism disperses the light into its color spectrum.

In the late 1800s, physicists were making accurate measurements of the
spectra of the
emissions from black bodies (objects which are opaque, or highly absorbing, to
the light they emit). Good examples of black bodies are the sun, the filament
of an incandescent lamp, and the burner of an electric stove. The color of a
black body depends on its temperature: A cool body emitting radiation of long
wavelengths, i.e., in the radio frequency range or in the infrared which are
invisible to the eye, a warmer body emitting radiation which includes shorter
wavelengths and appearing deep red, a still warmer body emitting radiation which
includes still shorter wavelengths and appearing yellow, and a hot body emitting
even shorter wavelengths and appearing white. The emissions are always over a
broad range of colors, or wavelengths, and their appearance is the net result of
seeing all of the colors at once. Examples of various blackbody spectra are
shown below. Computer simulations are given at
http://ephysics.physics.ucla.edu/physlets/eblackbody.htm.

Question: According to the above definition, is your body a
black body? Note: The human body can be seen in pitch darkness with thermal
imaging
goggles.

Classical physics could not explain the spectra of black bodies. It predicted
that the intensity (power emitted at a given wavelength) of emitted light should increase rapidly with decreasing
wavelength without limit (the "ultraviolet catastrophe"). In the
figure below, the curve labeled "Rayleigh-Jeans law" shows the
classically expected behavior.

However, the measured spectra actually showed an intensity maximum at a particular
wavelength, while the intensity decreased at wavelengths both above and below
the maximum. In order to explain the spectra, in 1900 the German physicist Max Planck
(1858 - 1947) was forced to make a desperate assumption for which he had
no physical explanation. As with classical physics, he assumed the body
consisted of vibrating oscillators (which were actually collections of atoms or
molecules).
However, in contrast to classical physics, which assumed that each oscillator
could absorb an arbitrary amount of energy from the radiation or emit an
arbitrary amount of energy to it, Planck was forced to assume that each
oscillator could receive or emit only discrete, quantized energies (E), such
that

E = hf
(Planck's formula)

where h (Planck's constant) is an exceedingly small number whose value we do not need here, and f is the frequency of vibration of the oscillator (the number
of times it vibrates per second). Each oscillator is assumed to vibrate only at a fixed frequency (although different oscillators in general had different
frequencies), so if it emitted some radiation, it would lose energy equal to hf,
and if it absorbed some radiation, it would gain energy equal to hf. Planck did
not understand how this could be, he merely made this empirical assumption in
order to explain the spectra. The figure above shows Planck’s prediction; this agreed with the measured spectra.

Also in the late 1800s, experimental physicists were measuring the emission
of electrons from metallic objects when they shined light on the object. This is
called the photoelectric effect. These experiments also could not be explained
using classical concepts. These physicists observed that emission of electrons
occurred only for light wavelengths shorter than a certain threshold value that depended on the metal. Classically, however, one expected that the emission
should not depend on wavelength at all, but only on intensity, with greater
intensities yielding more copious emission of electrons. A computer simulation
of the photoelectric effect is given at
http://phet.colorado.edu/simulations/sims.php?sim=Photoelectric_Effect (→Quantum
Phenomena→Photoelectric Effect). The diagram below illustrates the effect.

In one of a famous series of papers in 1905, Einstein explained the
photoelectric effect by starting with Planck’s concept of quantized energy
exchanges with light radiation, and making the startling assumption that these
quantized exchanges were a direct result of the quantization of light itself,
i.e. light consisted of discrete bundles of energy called photons, rather than
the continuous waves that had always been assumed in classical physics.
However, these bundles still had a wave nature, and could be characterized
by a wavelength, which determined their color. He also used Planck’s
relationship between energy and frequency (E = hf) to identify the energy of the photon,
and he used the relationship between velocity, frequency, and wavelength that
classical physics had always used (v=lf, where now
v=c= velocity of light). Einstein received the Nobel Prize for this
paper (not for his theories of relativity!).

In classical physics, the electromagnetic field connects charged particles to
each other (see Sections 2.4,
2.6). In
quantum physics, the force fields of classical physics are quantized, and the
quanta of the fields then become the force carriers. For example, photons are
the quanta of the electromagnetic field. In quantum physics, it is the
photons that connect charged particles to each other.

In addition to measuring the spectra of blackbody radiation in the 19th
century, experimental physicists also were familiar with the spectra emitted by
gases through which an electrical discharge (an electric current with enough
energy to strip some of the electrons from the atoms of the gas) was passing.
Examples of such discharges are the familiar neon sign, in which the gas is
neon; and the fluorescent light bulb, in which the gas is mercury vapor (the
fluorescent light bulb has special coatings on the inner walls which change the
spectrum of the light). The spectra of such light sources consist of emissions
at discrete, separated wavelengths, rather than over a continuous band of
wavelengths as in blackbody spectra. These spectra are called line spectra
because of their appearance when they are viewed with a spectrometer
(see
Section 3.1
and figure below). A
simulation applet of line spectra can be found at
http://jersey.uoregon.edu/vlab/elements/Elements.html.

Line spectra are another example of phenomena that could not be explained by
classical physics. Indeed, the explanation could not come until developments in
the understanding of the structure of atoms had been made by New Zealander physicist
Ernest Rutherford (1871 - 1937) and coworkers in 1911. By scattering alpha
particles (i.e., helium nuclei, which consist of two protons and two neutrons bound together) from thin gold foils, they discovered that the gold atom
consisted of a tiny (10-15 meters) very
dense, positively charged nucleus surrounded by a much larger (10-10
meters)
cloud of negatively charged electrons, see figure below. (Quantum mechanically, this picture is
not correct, but for now it is adequate.)

When classical physics was applied to such a model of the atom, it predicted
that the electrons could not remain in stable orbits about the nucleus, but
would radiate away all of their energy and fall into the nucleus, much as an
earth satellite falls into the earth when it loses its kinetic energy due to atmospheric
friction. In 1913, after Danish physicist Niels Bohr (1885 - 1962) had learned of these results, he constructed
a model of the atom that made use of the quantum ideas of Planck and Einstein.
He proposed that the electrons occupied discrete stable orbits without radiating
their energy. The discreteness was a result of the quantization of the orbits,
with each orbit corresponding to a specific quantized energy for the electron. The
electron was required to have a certain minimum quantum of energy corresponding
to a smallest orbit; thus, the quantum rules did not permit the electron to
fall into the nucleus. However, an electron could jump from a higher orbit to a
lower orbit and emit a photon in the process. The energy of the photon could
take on only the value corresponding to the difference between the energy of the
electron in the higher and lower orbits. An electron could also absorb a
photon and jump from a lower orbit to a higher orbit if the photon energy
equaled the difference in orbit energies, see figure below. Computer animations of the Bohr
model of photon emission and absorption in the hydrogen atom are given at
http://www.upscale.utoronto.ca/PVB/Harrison/BohrModel/Flash/BohrModel.html
and http://www.colorado.edu/physics/2000/index.pl (Table
of Contents→Science Trek Applets→Bohr's
Atom).

Bohr applied his theory to the simplest
atom, the hydrogen atom, which consists of one electron orbiting a nucleus of
one proton. The theory explained many of the properties of the observed line
spectrum of hydrogen, but could not explain the next more complicated atom, that
of helium, which has two electrons. Nevertheless, the theory contained the basic
idea of quantized orbits, which was retained in the more correct theories that
came later.

In the earliest days of the development of quantum theory, physicists, such
as Bohr, tried to create physical pictures of the atom in the same way they had
always created physical pictures in classical physics. However, although Bohr
developed his initial model of the hydrogen atom by using an easily visualized
model, it had features that were not understood, and it could not explain the
more complicated two-electron atom. The theoretical breakthroughs came when some
German physicists who were highly sophisticated mathematically, Werner
Heisenberg (1901 - 1976), Max Born (1882 - 1970), and
Pascual Jordan (1902 - 1980), largely abandoned physical pictures and created
purely
mathematical theories that explained the detailed features of the hydrogen
spectrum in terms of the energy levels and the intensities of the radiative transitions from one
level to another. The key feature of these theories was the use of matrices
instead of ordinary numbers to describe physical quantities such as energy,
position, and momentum. (A matrix is an array of numbers that obeys rules of
multiplication that are different from the rules obeyed by numbers.)

[Biographical notes: During World War II, Heisenberg worked on the German
nuclear energy project. Whether his role in the project was purely scientific or
whether he had political motives, either to work towards its success or towards
its failure, is still a matter of controversy. No such controversy exists over
the role of Jordan, who joined the Nazi party as a storm trooper in 1933, and
the Luftwaffe in 1939 as a weather analyst. Born, on the contrary, after being
classified as a Jew by the Nazis in 1933, left Germany and took a position at
the University of Cambridge, returning to Germany only after the War.]

The step of resorting to entirely mathematical theories that are not based
on physical pictures was a radical departure in the early days of quantum
theory, but today in developing the theories of elementary particles it is
standard practice. Such theories have become so arcane that physical pictures
have become difficult to create and to visualize, and they are usually developed to
fit the mathematics rather than fitting the mathematics to the picture.
Thus, adopting a positivist philosophy would prevent progress in developing
models of reality, and the models that are intuited are more mathematical than
physical.

Nevertheless, in the early 1920s some physicists continued to think in terms
of physical rather than mathematical models. In 1923, French physicist Louis de Broglie
(1892 - 1987) reasoned that
if light could behave like particles, then particles such as electrons could behave
like waves, and he deduced the formula for the wavelength of the waves:

l=h/p

where p is the momentum (mass x velocity) of the electron. Experiments
subsequently verified that electrons actually do behave like waves in
experiments that are designed to reveal wave nature. We will say more about such
experiments in Chapter 4. A computer demonstration of de Broglie waves is given at
http://www.colorado.edu/physics/2000/index.pl (Table
of Contents→Science Trek→de Broglie's
atom).

In physics, if there is a wave, there must be an equation that describes
how the wave propagates in time. De Broglie did not find it, but in 1926
Austrian-Irish physicist Erwin Schrödinger (1887- 1961) discovered the celebrated
equation that bears his name. The Schrödinger equation allows us to calculate precisely the
Schrödinger wave at all points in
space at any future time if we know the wave at all points in space at some
initial time. In this sense, even quantum theory is completely deterministic.

Schrödinger verified his
equation by using it to calculate the line emission spectrum from hydrogen,
which he could do without really understanding the significance of the waves. In
fact, Schrödinger misinterpreted the waves and thought they represented the
electrons themselves, see figure below. However, such an interpretation could not explain why
experiments always showed that the photons emitted by an atom were emitted at
random rather than predictable times, even though the average rate of emission
could be predicted from both Heisenberg’s and Schrödinger’s theories. It
also could not explain why, when an electron is detected, it always has a well-defined position in space, rather than being spread out over space like a wave.

The proper interpretation was discovered by German physicist Max Born (1882 -
1970) in 1926, who suggested
that the wave (actually, the absolute value squared of the amplitude or height of the wave, at
each point in space) represents the probability that the electron will
appear at that specified point in space if an experiment is done to measure
the location of the electron. Thus, the Schrödinger wave is a probability wave, not a wave that
carries force, energy, and momentum like the electromagnetic wave. Born's interpretation introduces two extremely
important features of quantum mechanics:

1) From the wave we can calculate
only probabilities, not certainties (the theory is probabilistic, not
deterministic).
2) The wave only tells us the probability of finding something
if we look, not what is there if we do not look. Quantum theory is not a
theory of objectively real matter (although Born thought the Schrödinger wave
was objectively real).

The first feature violates the second fundamental assumption of classical
physics (see
Section 2.2), i.e., that both the position and velocity of an object can be
measured with no limits on their precision except for those of the measuring
instruments. The second feature violates the first fundamental assumption of
classical physics, i.e., that the objective world
exists independently of any observations that are made on it.

Questions:
Suppose you accepted the principle that reality is probabilistic rather
than deterministic. How would it affect your notions of free will? How
would it affect your sense of control over your thoughts, feelings,
decisions, and actions? How would it affect your perceptions of other
people’s control over their thoughts, feelings, decisions, and actions? How would it affect
your judgments about yourself and others?

The terms "iron atom" and "electron" are heuristic attempts to
give names to the locations. However, this diagram in no way proves that there
are in reality such things as iron atoms and electrons. There is no way to prove
that (see
Section 1.1), but, by giving them names, we tend to be convinced that the
objects actually exist.

The probability measurements are represented by points so
densely packed that they appear to form surfaces rather than individual
measurements. The "iron atoms" are seen to be most probably located under the
blue peaks while the "electrons" are seen to be more diffusely located under the
circular rings. These are probability measurements of locations only,
not actual locations.

As Born proposed, quantum theory is intrinsically probabilistic in that in
most cases it cannot predict the results of individual observations. However, it
is deterministic in that it can exactly predict the probabilities that
specific results will be obtained. Another way to say this is that it can
predict exactly the average values of measured quantities, like position,
velocity, energy, or number of electrons detected per unit time in a beam of
electrons, when a
large number of measurements are made on identical electron beams. It cannot
predict the results of a single measurement. This
randomness is not a fault of the theory--it is an intrinsic property of nature.
Nature is not deterministic in the terms thought of in classical physics.

Another feature of the quantum world, the world of microscopic objects, is
that it is intrinsically impossible to measure simultaneously both the exact position
and momentum of a particle. This is the famous uncertainty principle of
Heisenberg, who derived it using the multiplication rules for the matrices that
he used for position and momentum. For example, an apparatus designed to measure
the position of an electron with a certain accuracy is shown in the following
diagram. The hole in the wall ensures that the positions of the electrons as
they pass through the hole are within the hole, not outside of it.

So far, this is not different from classical physics. However, quantum theory
says that if we know the position q of the electron to within an accuracy of Dq (the diameter of the hole), then our knowledge of the momentum
p (=mass x velocity) at that point
is limited to an accuracy Dp such that

(Dp)(Dq)>h
(Heisenberg uncertainty relation).

In other words, the more accurately we know the position of the electron (the
smaller Dq is), the less accurately we know the
momentum (the larger Dp is). Since momentum
is mass times velocity, the uncertainty in momentum is equivalent to an
uncertainty in velocity. The uncertainty in velocity is in the same direction as
the uncertainty in position. In the drawing above, the uncertainty in position
is a vertical uncertainty. This means that the uncertainty in velocity is also a
vertical uncertainty. This is represented by the lines
diverging (by an uncertain amount) after the electrons emerge from the hole
(uncertain vertical position) rather than remaining parallel as they are on the
left.

Likewise, an experiment designed to measure momentum with a certain accuracy
will not be able to locate the position of the particle with better accuracy
than the uncertainty relationship allows.

Notice that in the uncertainty
relationship, if the right side equals zero, then both Dp and Dq can
also be zero. This is the assumption of
classical physics, which says that if the particles follow parallel trajectories
on the left, they will not be disturbed by the hole, and they will follow
parallel trajectories on the right.

If we divide both sides of the uncertainty relation by the mass m of the
particle, we obtain

(Dv)(Dq)>h/m.

Here we see that the uncertainties in velocity v or position q are inversely
proportional to the mass of the particle. Hence, one way to make the right side
effectively zero is to make the mass very large. When numbers are put into this
relationship, it turns out that the uncertainties are significant when the
mass is microscopic, but for a macroscopic mass the uncertainty is unmeasurably
small. Thus, classical physics, which always dealt with macroscopic objects, was
close to being correct in assuming that the position and velocity of all objects
could be determined arbitrarily accurately.

The uncertainty principle can be understood from a wave picture. A wave
of
precisely determined momentum corresponds to an infinitely long train of
waves, all with the same wavelength, as is shown in the first of the
two wave patterns below. This wave is spread over all space, so its
location is indeterminate.

A wave of less precisely determined momentum can be obtained by superposing
(see Section 4.1) waves of slightly different wavelength (and therefore slightly different
momentum) together, as is shown in the second of the two patterns above. This results in a wave packet with a momentum spread Δp (uncertainty Δp), but
which is bunched together into a region of width Δx (uncertainty Δx ) instead of
being spread over all space.

The uncertainty relation is closely related to the complementarity principle,
which was first enunciated by Bohr. This principle states that
quantum objects (objects represented by quantum wave functions) have both a particle and a wave nature, and an attempt to measure
precisely a particle property will tend to leave the wave property
undefined, while an attempt to measure precisely a wave property will tend to
leave the particle property undefined. In other words, particle properties and
wave properties are complementary properties. Examples of particle properties
are momentum and position. Examples of wave properties are wavelength and
frequency. A precise measurement of momentum or position leaves wavelength
or frequency undefined, and a precise measurement of wavelength or frequency
leaves momentum or position undefined.

Question: Suppose the complementarity principle is extended to
macroscopic objects. For example, if your intent is to see a water wave,
you see a water wave but not a water particle. If your intent is to see
a water particle, you see a water particle but not a water wave. In
other words, you see only what you intend to see. Can you think of any
similar examples of this principle in your daily life?

We have seen that, even if the quantum wave function is objectively real, it
is a probability wave, not a physical wave. Furthermore, complementarity and uncertainty strongly imply that the electron (or any
other “particle”) exists neither as a physical particle nor a physical wave. But,
if so, in what form does it exist? So far, we have neglected the role of
the observer in all measurements. When we take the observer into account (see
Chapter 6), we shall see that quantum theory does not require physical particles
or waves (see
also
Section 1.1),
but it does require observations! We explore this provocative statement much further in later
chapters.

This page last updated October 1, 2010.
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